A coherent state undergoing a continuous nondemolition observation

20 August 2001

Physics Letters A 287 (2001) 19–22 www.elsevier.com/locate/pla

A coherent state undergoing a continuous nondemolition observation Przemysław Staszewski a,∗ , Gra˙zyna Staszewska b a Chair of Theoretical Foundations of Biomedical Sciences and Medical Informatics, Medical School, ul. Jagiello´ nska 13,

85-067 Bydgoszcz, Poland b Institute of Physics, Nicholas Copernicus University, ul. Grudzi¸ adzka 5, 87-100 Toru´n, Poland

Received 22 May 2000; received in revised form 15 June 2001; accepted 19 June 2001 Communicated by P.R. Holland

Abstract It is shown that there exists a continuous nondemolition observation (in Belavkin’s sense) preserving a coherent state of an open harmonic oscillator. The condition for such measurement, requiring a joint observation of position and momentum, is given.  2001 Elsevier Science B.V. All rights reserved. PACS: 03.65.Bz; 02.50.+s

1. Introduction The process of measurement, entailing an interaction of a quantum system in question with an apparatus, changes the state of the system. However, there is a possibility of introducing no additional distortion resulting from the reduction of the quantum state following the registered trajectory of the results of the measurement. This is the idea of nondemolition quantum measurements. A very important question is which properties of the observed quantum state are changed, and which are not. In particular, it is important to know whether the coherent state of a quantum harmonic oscillator can survive. We consider this problem in the framework

* Corresponding author.

E-mail addresses: [email protected] (P. Staszewski), [email protected] (G. Staszewska).

of the theory of continuous quantum measurement recently developed by Belavkin [1,2,4], cf. also [5]. This theory, based on his nondemolition principle, can be realized with the help of Bose fields interacting with the considered quantum system. The output fields of Gardiner and Collett [6], cf. also [7], serve for (indirect) nondemolition observation of the system. The time development of the posterior unnormalized wave function ϕ(t) ˆ (the wave function conditioned by the observation up to t) is given by Belavkin’s quantum filtering equation — a linear quantum stochastic differential equation (QSDE) of Itô type [8]. For the diffusion observation of one-dimensional quantum system S, Belavkin’s equation takes the following form: 
1 i ˆ ˆ dt + Lϕ(t) ˆ d Q(t), d ϕ(t) ˆ = − H + L† L ϕ(t) 2 h¯ ϕ(0) ˆ = ψ, (1.1) where ψ stands for the initial state, and H denotes the Hamiltonian of S.

0375-9601/01/$ – see front matter  2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 4 3 0 - 3

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P. Staszewski, G. Staszewska / Physics Letters A 287 (2001) 19–22

Eq. (1.1) describes the time-development of the posterior (pure) state of S, corresponding to the trajectory up to t of the observed output field coordinate process ˆ = 2 Re Bˆ † (t) = Bˆ † (t) + B(t), ˆ Q(t) (1.2) † ˆ ˆ where B (t) and B(t) are the Heisenberg evolutes of input field processes B † (t), B(t), related to creation and annihilation field operators b† (t), b(t) in the usual way: t B (t) = †

†





b (t ) dt ,

t B(t) =

0

b(t  ) dt  .

(1.3)

0

Due to the interaction of the system with the Bose reservoir (electromagnetic field) in the form i h¯ (L × dB † (t)/dt − L† dB(t)/dt), where L is the systematic coupling operator, the output process Qˆ carries nontrivial information about S. This information up to time t is stored in the wave function ϕ(t). ˆ The meaˆ surement of {Q(t), t
0} is self-nondemolition:   ˆ ˆ for all t
s
0, Q(s), Q(t) = 0, (1.4) and nondemolition: for any systematic operator Z, for all t
s
0,   ˆ ˆ Q(s), Z(t) = 0. (1.5) Thus, the continuous in time measurement of the field ˆ coordinate process {Q(t), t
0} does not introduce further distortion of motion of S, and Eq. (1.1) describes the time evolution of the state of S conditioned by q t — the trajectory up to t of the process ˆ {Q(t), t
0}. From the QSDE for Qˆ , ˆ = dQ(t) + 2 Re Lˆ † (t) dt, d Q(t)

(1.6)

one can see, that the measurement of Qˆ is an indirect and imperfect measurement of Re Lˆ † (t), the Heisenberg evolute of Re L† . In (1.6) Q(t) denotes the input field coordinate process, Q(t) = B † (t) + B(t), that the noise (d/dt)Q(t) plays a role of a random error. For L = 0, Eq. (1.1) reduces to the Schrödinger differential equation for the wave function of an unobserved quantum system. The multi-dimensional generalization of Eq. (1.1) is straightforward. In particular, the joint measurement of noncommuting observables is possible in this approach, provided that the couplings enabling measurements of noncommuting components to independent field modes are introduced.

In our recent paper [9], we have analyzed the solution to the problem of the time-evolution of a Gaussian wave packet of quantum particle moving in the field of a linear force F (x) = h¯ γ x with continuously observed position. For the case of a harmonic oscillator (γ < 0) we have shown that for the initial coherent state the asymptotic position dispersion is always smaller than the initial one (while for the unobserved particle initially prepared in a coherent state the position dispersion is constant). Similar analysis shows that also for a harmonic oscillator with continuously observed momentum the coherent state is not preserved. As it is known, a coherent state of a harmonic oscillator is the only Gaussian wave packet invariant under the Schrödinger evolution. A measurement preserving a coherent state of a harmonic oscillator was constructed in [1] by means of quantum linear filtering theory. In this Letter, we discuss the possibility of preserving a coherent state of a harmonic oscillator undergoing a continuous joint nondemolition observation of its position and momentum in the framework of the general quantum stochastic scheme of continuous nondemolition observation. The detailed presentation of the time development of the Gaussian wave packet in this case will be published elsewhere [10].

2. The observation of a coherent state Let us recall the problem of a quantum particle moving in the field of a linear force with continuously observed (a) position [11–13], (b) momentum [14,15], (c) position and momentum [1,10,14,15]. To realize the position measurement, the systematic coupling operator L has √ to be proportional to the position operator X, L = λ/2 X. Similarly, for the momentum measurement one has to specify the system√ atic coupling operator in the form δ/2 P , where P stands for the momentum operator. For the case of the joint measurement of position and momentum L is a vector-operator,    L = (L1 , L2 ) = λ/2 X, δ/2 P , (2.1) coupling the considered harmonic oscillator to independent modes of the Bose field. Consequently,

P. Staszewski, G. Staszewska / Physics Letters A 287 (2001) 19–22

a generalization of the filtering equation (1.1) to the case of two-dimensional observation has to be considered [15]:  2 1 † i d ϕ(t) ˆ =− H + ˆ dt Lk Lk ϕ(t) h¯ 2 k=1

+

2

Lk ϕ(t) ˆ d Qˆ k (t).

(2.2)

k=1

Note, that according to QSDE (1.6) the positive constants λ and δ are interpreted as the accuracies of the position and momentum measurements, respectively. Let us recall that for all the cases (a)–(c) of obˆ x) = ϕ(t, servation the normalized solution ψ(t, ˆ x) × ( ϕ(t, ˆ x) )−1 to Belavkin’s quantum filtering equation corresponding to the initial state ψ(x) of the form of Gaussian wave packet,

  1 i 2 −1/4 2 ψ(x) = 2πσx exp − 2 (x − q) + px , h¯ 4σx (2.3) has the same form

 2 i m ˆ ψ(t, x) = cˆ exp ω(t) x − q(t) ˆ + p(t)x ˆ . 2h¯ h¯ (2.4) In (2.4) q(t), ˆ p(t) ˆ are posterior mean values (i.e., ˆ mean values calculated with respect to ψ(x, t)), which fulfill linear filtration equations (Hamilton–Langevin equations), cˆ = (2τq2 π)−1/4 up to inessential stochastic phase factor, and τ 2 = q2 − qˆ 2 is the posterior po-

for the case (b) i h¯ γ d ω(t) = + (h¯ δm + i)ω2 (t), dt m for the case (c) h¯ d ω(t) = − (λ − iγ ) + (h¯ δm + i)ω2 (t), dt m with the initial condition ω(0) = ω0 = −

τx2 (t) = −

h¯ , 2m Re ω(t)

τp2 (t) = −

h¯ m|ω(t)|2 . 2 Re ω(t)

(2.5)

As it is shown in [11,14], the function ω(t) satisfies the Riccati differential equation taking, for the considered case of observation (a)–(c), the form for the case (a) h¯ d ω(t) = − (λ − iγ ) + iω2 (t), dt m

(2.6)

(2.7)

(2.8)

(2.9)

Let us now consider the case of the Gaussian wave packet with dispersion constant in time, i.e., satisfying the condition ω(t) = ω0 . To this end we put dω/dt = 0 in Eqs. (2.6)–(2.8). We get, respectively,
 hλ h¯ γ ¯ i (2.10) + ω02 = , m m
 h¯ γ i (2.11) + ω02 = −h¯ δmω02 , m
 h¯ γ hλ ¯ + ω02 = − h¯ δmω02 . i (2.12) m m Let us first notice that for the case of unobserved particle, i.e., for λ = 0, δ = 0, the state with constant in time dispersion can occur only for negative γ (γ = −|γ |), i.e., for a harmonic oscillator, and the condition h¯ |γ | (2.13) m has to be satisfied. Due to (2.9) condition (2.13) can be also written in the form

ω02 =

q

sition dispersion. The posterior position and momentum dispersions are given in terms of ω(t) by the formulas

h¯ . 2mσx2

21

|γ | =

h¯ . 4mσx4

(2.14)

For the observed particle, the condition (2.10) as well as (2.11) cannot be satisfied for positive λ and δ, respectively, because the left-hand side is purely imaginary while the right-hand side of Eq. (2.10) or (2.11) is a real number. Thus, the coherent state of the harmonic oscillator cannot be preserved under the nondemolition observation of position and momentum taken separately. However, when the joint observation of position and momentum is performed, condition (2.12) can be satisfied for nonzero λ and δ. Similarly to the case of Eqs. (2.10), (2.11), equality (2.12) can be fulfilled only if both sides of (2.12) are equal to 0. Therefore,

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P. Staszewski, G. Staszewska / Physics Letters A 287 (2001) 19–22

conditions (2.13) and λ = δm2 ω02

Acknowledgement (2.15)

have to be satisfied. Using (2.13) condition (2.15) can be rewritten in the form λ = h¯ mδ|γ |.

This work was partially supported by the BA Grant No. BW 34/2001.

(2.16)

Thus, the coherent state of a harmonic oscillator is preserved provided the joint observation of its position and momentum is taken in such a way that condition (2.16) is satisfied. In view of the form (2.1) of the coupling operator for the joint position-momentum measurement, condition (2.16) means that Hermitian and anti-Hermitian parts of an annihilation operator a are simultaneously observed with the same accuracy µ by means of independent field processes. Therefore, choosing    L = (L1 , L2 ) = µ/2 a, −i µ/2 a (2.17) † a + 1/2) one gets Eq. (2.2) in and putting H = hω(a ¯ the form 

 1 µ † † ˆ dt d ϕ(t) ˆ = − iω a a + + a a ϕ(t) 2 2  ˆ + 2µ a ϕ(t) (2.18) ˆ d Q(t),

ˆ ˆ 2 (t). (For more details on where Q(t) = Qˆ 1 (t) + i Q the complex diffusion observation see [3].) Again, for initial state (2.3), the normalized solution to Eq. (2.18) has the form (2.4). The corresponding Riccati differential equation for the function ω(t) reads

 1/2 m d ω(t) = µ + i ω2 (t) dt 4h¯ |γ |

 h¯ m|γ | 1/2 − (2.19) µ − iγ . m 4h¯ One check easily that for the initial condition ω(0) = ω0 with ω0 given by equality (2.13) the function ω is constant in time, that is, the coherent state is preserved.

References [1] V.P. Belavkin, in: A. Blaquiere, S. Diner, G. Lochak (Eds.), Information Complexity and Control in Quantum Physics, Springer, Berlin, 1987, p. 311. [2] V.P. Belavkin, in: A. Blaquiere (Ed.), Modeling and Control of Systems in Engineering, Quantum Mechanics, Economics and Biosciences, Springer, Berlin, 1988, p. 245. [3] A. Barchielli, V.P. Belavkin, J. Phys. A: Math. Gen. 24 (1991) 1495. [4] V.P. Belavkin, Found. Phys. 24 (1994) 685; P. Staszewski, Quantum Mechanics of Continuously Observed Systems, N. Copernicus University Press, Toru´n, 1993, and references therein. [5] L. Diósi, Phys. Lett. A 129 (1988) 419; L. Diósi, Phys. Lett. A 132 (1988) 233; A.C. Doherty, S.M. Tan, A.S. Parkins, D.F. Walls, Phys. Rev. A 60, 2380. [6] C.W. Gardiner, M.J. Collett, Phys. Rev. A 31 (1985) 3761. [7] A. Barchielli, Phys. Rev. A 34 (1986) 1642. [8] R.L. Hudson, K.R. Parthasarathy, Comm. Math. Phys. 93 (1984) 301. [9] P. Staszewski, G. Staszewska, Open Syst. Inf. Dyn. 7 (2000) 77. [10] P. Staszewski, G. Staszewska, A. D¸abrowska, A quantum particle undergoing a continuous nondemolition measurement, in preparation. [11] V.P. Belavkin, P. Staszewski, Phys. Lett. A 140 (1989) 359. [12] P. Staszewski, G. Staszewska, Open Syst. Inf. Dyn. 5 (1998) 391. [13] P. Staszewski, G. Staszewska, Phys. Scr. 62 (2000) 117. [14] D. Chru´sci´nski, P. Staszewski, Phys. Scr. 45 (1992) 193. Eqs. (2.13) and (3.8) of this reference are Eqs. (2.5) and (2.6) of the present Letter. The factor 3 appearing in the coefficient against ω2 in (3.8) is mistakenly given and should be omitted. [15] P. Staszewski, in: V.P. Belavkin, O. Hirota, R.L. Hudson (Eds.), Quantum Communications and Measurement, Plenum Press, New York, 1995, p. 119.